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Math Help - Finding the Line Integral

  1. #1
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    Finding the Line Integral

    Hi
    I am having trouble on working out the following question:

    Given C is the anticlockwise circular path x^2+y^2=4, starting and ending at (0,2) evaluate the line integrals \oint_C F.dr where F = xyi+xyj

    This is what i have done

    r=xi + yj
    dr = i + j

    x=2cos(\theta), y=2sin(\theta)

    F = (4cos(\theta) sin(\theta))i + (4cos\theta sin\theta )j

    \int_{0}^(2\pi) (4cos\theta sin\theta)i + (4cos\theta sin\theta )j \cdot i + j

    P.S
    Last edited by Paymemoney; May 27th 2011 at 05:03 PM.
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  2. #2
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    dr\neq \hat{\imath} + \hat{\jmath}
    dr = dx\hat{\imath} + dy \hat{\jmath}

    If
    x=2\cos\theta, y=2\sin\theta

    what are dx and dy?
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  3. #3
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    ok i have attempted this question again and this is what i have done but this is incorrect.


    This is what i have have done:
    \int_0^{2\pi} 2cos(\theta)+2cos(\theta)2sin(\theta)*2cos(\theta)-2sin(\theta) d(\theta)

    \int_0^{2\pi} 8cos(\theta)^2 * sin(\theta) - 8cos(\theta) * sin(\theta)^2 + 8cos(\theta)^2 * sin(\theta) - 8cos(\theta)* sin(\theta)^2

    \int_0^{2\pi} 8(cos(\theta)^2 * sin(\theta) - cos(\theta) * sin(\theta)^2 + cos(\theta)^2 * sin(\theta) - cos(\theta)* sin(\theta)^2)
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  4. #4
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    Without parentheses, it is almost impossible to read what you have written. With x= 2 cos(\theta), y= 2 sin(\theta), dx= -2 sin(\theta)dt and dy= 2 cos(\theta)dt. F(x,y) becomes 4sin(\theta)cos(\theta)i+ 4 sin(\theta)cos(\theta)j and so the integral is
    8\int_{\pi/2}^{5\pi/2} (-sin(\theta)cos^2(\theta)+ sin^2(\theta)cos(\theta))dt
    Personally, I would have used x= 2sin(\theta),  y= 2cos(\theta) so I could integrate from 0 to 2\pi.
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